First slide

Gravitaional potential energy

Question

Escape velocity on the surface of a planet of radius R and density $ρ$ is V. What will be the escape velocity on the surface of a planet of radius 2R and density 4 $ρ$ ?

Moderate

A

V
B

$\frac{V}{2}$
C

4V
D

2V

Solution

(3)
$V_{e} = \sqrt{\frac{2 G M}{R}} = \sqrt{\frac{2 G . \frac{4}{3} π R^{3} ρ}{R}} = R \sqrt{\frac{8}{3} π G . ρ}$
$\begin{array}{l} ∴ \frac{V^{'} e}{V} = \frac{R^{'}}{R} . \sqrt{\frac{ρ^{'}}{ρ}} = \frac{2 R}{R} . \sqrt{\frac{4 ρ}{ρ}} = 4. \\ \Rightarrow V^{'} e = 4 V . \end{array}$

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